Cosmological singularity theorems and some modified gravity theories

In the first part of the present talk, the cosmological singularity theorems of Hawking will be reviewed. In the second part, it will be considered the possibility that the specific energy conditions for matter fields that these theorems assume may be violated. However, some generalizations of the Hawking theorems will be found even in these situations. In particular, the Hawking theorems will be generalized for F(R) theories formulated in the metric formalism. Certain results of Ringstrom about scalar fields in GR will be very important for obtaining such results.